Cardinal Number

In Mathematics , cardinal numbers, or '''cardinals''' for short, are a generalized kind of Number used to denote the size of a Set , known as its ''' Cardinality '''. For Finite Set s the cardinality is given by a Natural Number , being simply the number of elements in the set. There are also '' Transfinite '' cardinal numbers to describe the sizes of infinite sets. On the one hand, a Proper Subset ''A'' of an Infinite Set ''S'' may have the same cardinality as ''S''. On the other hand, perhaps also counterintuitively, not all infinite sets have the same cardinality. There is a formal characterization that explains how some infinite sets have cardinalities that are strictly smaller than other infinite sets. The cardinal numbers are: $0, 1, 2, 3, \cdots n, \cdots ; \aleph_0, \aleph_1, \aleph_2, \cdots \aleph_{\alpha}, \cdots$ . That is, they are the Natural Number s (finite cardinals) followed by the Aleph Number s (infinite cardinals). The aleph numbers are indexed by Ordinal Number s. The natural numbers and aleph numbers are subclasses of the ordinal numbers. If the Axiom Of Choice fails, the situation becomes more complicated — there are additional infinite cardinals which are not alephs. Concepts of cardinality are embedded in most branches of mathematics and are essential to their study. Cardinality is also an area studied for its own sake as part of Set Theory , particularly in trying to describe the properties of Large Cardinal s. HISTORY The notion of cardinality, as now understood, was formulated by Georg Cantor , the originator of Set Theory , in 1874–1884. Cantor first established cardinality as an instrument to compare finite sets; e.g. the sets {1,2,3} and {2,3,4} are not ''equal'', but have the ''same cardinality'', namely three. Cantor identified the fact that One-to-one Correspondence is the way to tell that two sets have the same size, called "cardinality", in the case of finite sets. Using this one-to-one correspondence, he applied the concept to infinite sets; e.g. the set of natural numbers N = {0, 1, 2, 3, ...}. He called these cardinal numbers Transfinite Cardinal Numbers , and defined all sets having a one-to-one correspondence with N to be Denumerably Infinite Sets . Naming this cardinal number $\aleph_0$ , Aleph-null , Cantor proved that any unbounded subset of N has the same cardinality as N, even if this might appear at first to run contrary to intuition. He also proved that the set of all Ordered Pair s of natural numbers is denumerably infinite (which implies that the set of all Rational Number s is denumerable), and later proved that the set of all Algebraic Number s is also denumerably infinite. Each algebraic number ''z'' may be encoded as a finite sequence of integers which are the coefficients in the polynomial equation of which it is the solution, i.e. the ordered n-tuple $(a_0, a_1, ..., a_n),\; a_i \in \mathbb{Z},$ together with a pair of rationals $(b_0, b_1)$ such that ''z'' is the unique root of the polynomial with coefficients $(a_0, a_1, ..., a_n)$ that lies in the interval $(b_0, b_1)$ . In his 1874 paper, Cantor proved that there exist higher-order cardinal numbers by showing that the set of real numbers has cardinality greater than that of N. His Original Presentation used a complex argument with Nested Intervals , but in an 1891 paper he proved the same result using his ingenious but simple Diagonal Argument . This new cardinal number, called the Cardinality Of The Continuum , was termed ''c'' by Cantor. Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ( $\aleph_0$ , aleph-null) and that for every cardinal number, there is a next-larger cardinal $(\aleph_1, \aleph_2, \aleph_3, \cdots)$ . His Continuum Hypothesis is the proposition that ''c'' is the same as $\aleph_1$ , but this has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved under the standard assumptions. MOTIVATION In informal use, a cardinal number is what is normally referred to as a '''counting number'''. They may be identified with the Natural Numbers beginning with 0 (i.e. 0, 1, 2, ...). The counting numbers are exactly what can be defined formally as the Finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic. More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set which has exactly the right size, e.g. 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c} which has 3 elements. However when dealing with Infinite Set s it is essential to distinguish between the two — the two notions are in fact different for infinite sets. Considering the position aspect leads to ordinal numbers, while the size aspect is generalized by the cardinal numbers described here. The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions. A set ''Y'' is at least as big as, or greater than or equal to a set ''X'' if there is an Injective (one-to-one) Mapping from the elements of ''X'' to the elements of ''Y''. A one-to-one mapping identifies each element of the set ''X'' with a unique element of the set ''Y''. This is most easily understood by an example; suppose we have the sets ''X'' = {1,2,3} and ''Y'' = {a,b,c,d}, then using this notion of size we would observe that there is a mapping: : 1 → a : 2 → b : 3 → c which is one-to-one, and hence conclude that ''Y'' has cardinality greater than or equal to ''X''. Note the element d has no element mapping to it, but this is permitted as we only require a one-to-one mapping, and not necessarily a one-to-one and Onto mapping. The advantage of this notion is that it can be extended to infinite sets. We can then extend this to an equality-style relation. Two Set s ''X'' and ''Y'' are said to have the same cardinality if there exists a Bijection between ''X'' and ''Y''. By the Schroeder-Bernstein Theorem , this is equivalent to there being ''both'' a one-to-one mapping from ''X'' to ''Y'' ''and'' a one-to-one mapping from ''Y'' to ''X''.
	Formally, The Order Among Cardinal Numbers Is Defined As Follows: &nbsp''X''&nbsp ≤ &nbsp''Y''&nbsp Means That There Exists An	"http://wwwinformationdelightinfo/information/entry/injective" class="copylinks">Injective function from ''X'' to ''Y'' The Cantor–Bernstein–Schroeder Theorem states that if &nbsp''X''&nbsp ≤ &nbsp''Y''&nbsp and &nbsp''Y''&nbsp ≤ &nbsp''X''&nbsp then &nbsp''X''&nbsp = &nbsp''Y''&nbsp The Axiom Of Choice is equivalent to the statement that given two sets ''X'' and ''Y'', either &nbsp''X''&nbsp ≤ &nbsp''Y''&nbsp or &nbsp''Y''&nbsp ≤ &nbsp''X''&nbsp
	A Set ''X'' Is	"http://wwwinformationdelightinfo/information/entry/Dedekind-infinite" class="copylinks">Dedekind-infinite if there exists a Proper Subset ''Y'' of ''X'' with &nbsp''X''&nbsp = &nbsp''Y''&nbsp, and Dedekind-finite if such a subset doesn't exist The Finite cardinals are just the Natural Numbers , ie, a set ''X'' is finite if and only if &nbsp''X''&nbsp = &nbsp''n''&nbsp = ''n'' for some natural number ''n'' Any other set is Infinite Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones It can also be proved that the cardinal <math>\aleph_0</math> (aleph-0, where aleph is the first letter in the Hebrew Alphabet , represented <math>\aleph</math>) of the set of natural numbers is the smallest infinite cardinal, ie that any infinite set has a subset of cardinality <math>\aleph_0</math> The next larger cardinal is denoted by <math>\aleph_1</math> and so on For every Ordinal α there is a cardinal number <math>\aleph_{\alpha},</math> and this list exhausts all infinite cardinal numbers
	:<math>X + Y	X \cup Y</math>
	:<math>X\cdotY	X imes Y</math>

Note That 2<sup>&nbsp''X''&nbsp</sup> Is The Cardinality Of The "http://wwwinformationdelightinfo/information/entry/power_set" class="copylinks">Power Set of the set ''X'' and Cantor's Diagonal Argument shows that 2<sup>&nbsp''X''&nbsp</sup> > &nbsp''X''&nbsp for any set ''X'' This proves that there exists no largest cardinal In fact, the Class of cardinals is a Proper Class

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Cardinal Number